# Hecke algebra

Let $f$ be a modular form^{} for $\mathrm{\Gamma}$ a congruence subgroup of ${\text{SL}}_{2}(\mathbb{Z})$.

$$f(z)=\sum _{n=0}^{\infty}{a}_{n}{q}^{n}$$ | (1) |

where $q={e}^{2i\pi \tau}$.

For $m\in \mathbb{N}$, let ${T}_{m}f(z)=\sum _{n=0}^{\infty}{b}_{n}{q}^{n}$ with :

$${b}_{n}=\sum _{d|\mathrm{gcd}(m,n)}^{}{d}^{k-1}{a}_{mn/{d}^{2}}$$ | (2) |

In particular, for $p$ a prime, ${T}_{p}f(z)=\sum _{n=0}^{\infty}{b}_{n}{q}^{n}$ with;

$${b}_{n}={a}_{pn}+{p}^{k-1}{a}_{n/p}$$ | (3) |

where ${a}_{n/p}=0$ if $n$ is not divisible by $p$.

The operator ${T}_{n}$ is a linear operator on the space of modular forms called a *Hecke operator ^{}*.

The Hecke operators leave the space of modular forms and cusp forms^{} invariant
and turn out to be self-adjoint^{} for a scalar product^{} called the Petersson
scalar product. In particular they have real eigenvalues^{}. Hecke operators
also satisfy multiplicative properties that are best summarized by the formal
identity^{}:

$$\sum _{n=1}^{\infty}{T}_{n}{n}^{-s}=\prod _{\mathit{p}}{(1-{T}_{p}{p}^{-s}+{p}^{k-1-2s})}^{-1}$$ | (4) |

That equation in particular implies that ${T}_{m}{T}_{n}={T}_{n}{T}_{m}$ whenever $\mathrm{gcd}(n,m)=1$.

The set of all Hecke operators is usually denoted $\mathbb{T}$ and is called the *Hecke algebra*.

## 0.1 Group algebra example

Definition 0.1
Let ${G}_{lcd}$ be a locally compact totally disconnected group; then the
*Hecke algebra $\mathrm{H}\mathit{}\mathrm{(}{G}_{l\mathit{}c\mathit{}d}\mathrm{)}$ of the group* ${G}_{lcd}$ is defined as the convolution algebra of
locally constant complex-valued functions on ${G}_{lcd}$ with compact support.

Such $\mathscr{H}(G)$ algebras^{} play an important role in the theory of
decomposition of group representations^{} into tensor products^{}.

Title | Hecke algebra |
---|---|

Canonical name | HeckeAlgebra |

Date of creation | 2013-03-22 14:08:13 |

Last modified on | 2013-03-22 14:08:13 |

Owner | olivierfouquetx (2421) |

Last modified by | olivierfouquetx (2421) |

Numerical id | 13 |

Author | olivierfouquetx (2421) |

Entry type | Definition |

Classification | msc 11F11 |

Classification | msc 20C08 |

Related topic | ModularForms |

Related topic | AlgebraicNumberTheory |

Defines | Hecke operator |

Defines | Hecke algebra $H(G)$ of the group G |